The M\"obius function of the small Ree groups

TL;DR

This paper computes the M"obius function for small Ree groups, revealing their subgroup structure and applications in counting generating sets, dessins d'enfants, and probabilistic group generation.

Contribution

It determines the M"obius function of small Ree groups and describes their maximal subgroups using permutation representations, advancing understanding of their algebraic and combinatorial properties.

Findings

Calculated the M"obius function for small Ree groups.

Described maximal subgroups via permutation representations.

Applied results to counting epimorphisms and probabilistic generation.

Abstract

The M\"obius function for a group, $G$, was introduced in 1936 by Hall in order to count ordered generating sets of $G$. In this paper we determine the M\"obius function of the simple small Ree groups, $R(q)={}^2G_2(q)$ where $q=3^{2m+1}$ for $m>0$, using their 2-transitive permutation representation of degree $q^3+1$ and describe their maximal subgroups in terms of this representation. We then use this to determine $\vert$Epi$(\Gamma,G)\vert$ for various $\Gamma$, such as $F_2$ or the modular group $PSL_2(\mathbb{Z})$, with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.